BRST Cohomology and Its Application to QED

We construct the BRST cohomology under a positive definite inner product and obtain the Hodge decomposition theorem at a non-degenerate state vector space $V$. The harmonic states isomorphic with a BRST cohomology class correspond to the physical Hilbert space with positive norm as long as the completeness of $Q_{BRST}$ is satisfied. We explicitly define ``co-BRST'' operator and analyze the quartet mechanism in QED.